Iterative fisher linear discriminant analysis

ABSTRACT

An exemplary method includes receiving an image data set that comprises a multidimensional property space and data classifiable into data classes, determining a projection vector for data of the data set wherein the projection vector maximizes a ratio of between-class scatter to within-class scatter, selecting a reference for the vector, projecting at least some of the data onto the vector, measuring distances from the reference to at least some of the data, classifying at least some of the data into data classes based on a nesting analysis of the distances, eliminating the classified data from the image data set to produce a modified image data set and deciding whether to determine another projection vector for the modified image data set. Various other exemplary methods, devices, systems, etc., are also disclosed.

RELATED APPLICATIONS

This application claims the benefit of a U.S. Provisional Applicationentitled “Iterative Fisher Linear Discriminant Analysis”, filed Feb. 17,2004, to Trifonov and Lalyko, having Provisional Application Ser. No.60/545,652,, which is incorporated by reference herein.

FIELD OF INVENTION

Subject matter disclosed herein pertains to classifiers and, inparticular, to classification schemes for classifying image data.

BACKGROUND

Data classification problems are commonly encountered in the technicalarts. Examples include determining if tumors are malignant or benign,deciding if an article of manufacture is within tolerance or not,establishing the degree to which a combination of medical tests predictsa disease, classifying the content of an image, determining therelevance or irrelevance of information, and so on.

Given examples of classes or categories, each associated with multipleattributes or properties, the task is to determine regions of attributespace that define the classes. This makes it possible to subsequentlycategorize newly acquired data into classes based on the values of theattributes or properties of the data when the class membership is notknown in advance.

Important aspects of the data as regards ease of classification includethe number of classes contained in the data, the number of attributesfor each datum, i.e., the dimensionality of the property space, and thenature of the distribution of classes within the property space. Manymethods of classification are available. A number of the most useful arereviewed and compared in “A Comparison of Prediction Accuracy,Complexity and Training Time of Thirty-three Old and New ClassificationAlgorithms”, T.-S. Lim, W.-S. Loh and Y.-S. Shih, Machine Learning, v.40, p. 203-229, 2000.

Four important characteristics of a classifier are the accuracy ofclassification, the training time required to achieve classification,how that training time scales with the number or classes and thedimensionality of the property space describing the data, and howconsistent or robust is the performance of the classifier acrossdifferent data sets.

One well-established method of classification is linear Fisherdiscriminant analysis, which is notable for an especially favorablecombination of good classification accuracy coupled with consistencyacross different data sets and a low training time. The last isespecially important where classification must occur in real-time ornearly so.

Fisher discriminant analysis defines directions in property space alongwhich simultaneously the between-class variance is maximized and thewithin-class variance is minimized. In other words, directions inproperty space are sought which separate the class centers as widely aspossible while simultaneously representing each class as compactly aspossible. When there are two classes there is a single discriminantdirection.

Depending on the dimensionality of the property space, a line, plane orhyperplane constructed normal to this direction may be used to separatethe data into classes. The choice of the location of the plane (or itsequivalent) along the discriminant coordinate depends on theclassification task. For example, the location may be chosen to providean equal error for classification of both classes. As another example,the location may be chosen to maximize the probability that allinstances of a given class are correctly detected without regard tofalse positive identification of the remaining class. When there aremore than two classes Fisher discriminant analysis provides a family ofdiscriminant direction vectors, one fewer in number than the number ofclasses. Planes can be positioned along these vectors to pairwiseseparate classes.

A concept related to Fisher discriminant analysis is principal componentanalysis, otherwise known as the Karhunen-Loeve transform. Its purposeis to transform the coordinates of a multi-dimensional property space soas to maximize the variance of the data along one of the newcoordinates, which is the principal component. Unlike Fisherdiscriminant analysis, the objective is to determine a direction thatmaximizes the overall variance of the data without regard to thevariance within classes. As a result of the transform, initialorthogonal property vectors become resulting orthogonal principalcomponent vectors by rotation. In contrast, however, discriminantvectors are not, in general, orthogonal, having their directionsdetermined by the distribution of class properties. Thus, the vectorsdefining, on the one hand, the discriminant directions and, on theother, the principal component directions are in general distinct andnon-coincident.

Underlying the linear Fisher discriminant analysis is the idea thatclasses within the data have properties that are normally distributed,i.e. each property has a Gaussian distribution about a mean value. Tothe extent that the actual property distributions of the data violatethis assumption the performance of this classifier degrades. That isespecially the case when the distributions are multi-modal, i.e., when agiven class is represented my multiple groups or clusters of propertiesthat are well-separated within property space and interspersed withsimilar clusters representing other classes.

An alternative view of this problem is that a plane (or its equivalent)positioned normal to a discriminant direction is an insufficientlyflexible entity to describe the boundary between modes or clusterswithin property space. The difficulty is readily appreciated with asimple example. Given two classes in a two dimensional property plane,if the class distributions lie on a line such that a single propertydistribution for class 1 is flanked on either side by distributions forclass 2, no single straight line will completely separate the two setsof property distributions.

It is to cope with problems such as this that a wealth of variousclassifiers has been devised. For example, one technique imposes aclassification tree on the data using discriminant analysis to determinethe branching at each level of the tree (see “Split Selection Methodsfor Classification Trees”, W.-Y. Loh and Y.-S. Shih, Statistica Sinica,v. 7, p. 815-840, 1997). However, the optimal estimation of the treerequires considerable extra computation and the method is more than anorder of magnitude slower than simple linear Fisher discriminantanalysis.

In view of the fact that very few classifiers combine the speed andaccuracy of linear discriminant analysis, there is a need to improve theclassification accuracy of this classifier for data with complexmulti-modal attribute distributions while maintaining a minimal impacton the classification time. Various exemplary methods, devices, systems,etc., disclosed herein aim to address this need and/or other needspertaining to classification of data such as image data.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A and 1B illustrate two scenarios for classification of atwo-class model.

FIGS. 2A and 2B illustrate exemplary techniques for projecting data andbinning information.

FIG. 3 illustrates an exemplary nesting analysis and exemplaryrelationships for binned information.

FIG. 4 illustrates an exemplary nesting analysis of binned distanceinformation.

FIG. 5 illustrates an exemplary scenario comparing linear discriminantanalysis and principle component analysis. FIG. 6 illustrates anexemplary nesting analysis for a generic multiclass scenario.

FIG. 7 illustrates an exemplary method for classifying data.

FIG. 8 illustrates an exemplary method for classifying data using one ormore types of analyses (e.g., linear discriminant analysis, principlecomponent analysis, etc.).

FIGS. 9A-D are images originally in color and presented in grayscalewherein the image of FIG. 9D is the result of an exemplary iterativeclassification technique and wherein the image of FIG. 9C is the resultof a conventional classification technique.

FIGS. 10A-C are images originally in color and presented in grayscalewherein the image of FIG. 10C is the result of an exemplary iterativeclassification technique.

FIG. 11 illustrates an example computing system capable of implementingthe various exemplary methods described herein.

DETAILED DESCRIPTION

Described herein are various exemplary methods, devices, systems, etc.,that provide an iterative classifier based on linear Fisher discriminantanalysis. For example, in one implementation, a classifier is providedin which property distances are projected onto a discriminant vector toform property histograms. Classification may rely on such propertyhistograms wherein a nesting analysis has the objective of identifyingdata within classes and possibly eliminating correctly classified data.Where correctly classified data are eliminated, remaining data may bereclassified using a subsequent linear Fisher discriminant analysis. Theprocess may be iterated until full classification is achieved or nofurther simplification is possible.

Various exemplary methods, devices, systems, etc., optionally includeprojection of data onto more than one projection vector and subsequentformation of one or more property histograms. For example, oneprojection vector may arise from a linear discriminant analysis whileanother projection vector may arise from a principle component analysis.In this example, a histogram or histograms may be formed for eachprojection vector and a decision made as to which vector allows for moreaccurate discrimination between classes and hence classification ofdata. A criterion for this decision may consider which approach resultsin a maximal simplification of the data. A nesting analysis of histograminformation may be employed for decision making and/or classification ofdata. Correctly classified data may be identified and eliminated, asappropriate, and the remaining data subject to reclassification usingone or both of the projection analyses. The process is may be iterateduntil full classification is achieved or no further simplification ispossible.

Graphical Representation of a Two-class Model

FIGS. 1A and 1B show graphical representations of two scenarios of atwo-class model. In a first scenario, as illustrated in FIG. 1A, a plot105 of a first property versus a second property (e.g., a propertyspace) includes data within a first class and data within a secondclass. One direction 107 in the property space is associated with ananalysis that aims to maximize variance between classes and minimizevariance within classes while another direction 109 is associated withan analysis that aims to maximize overall variance. In higher dimensionproperty spaces, such classes may be defined using more than twodimensions. Consider a three-dimensional space wherein a class may bedefinable by a “volume”. Extension beyond a two-dimensional propertyspace is discussed further below.

In this scenario, upon projection of the class data onto the direction107, the classification ability of the analysis resulting in thedirection 107 is quite limited due to the relationship of the classes inthe property space. Upon projection of the class data onto the direction109, the classification ability of the analysis resulting in thedirection 109 has arguably more potential. Thus, the scenario of plot105 demonstrates an instance where an analysis that seeks to maximizeoverall variance may provide useful information for classifying data ina property space. Further, the scenario of plot 105 demonstrates aninstance where an analysis that seeks to maximize variance betweenclasses and minimize variance within classes may be limited inusefulness. In a second scenario, as illustrated in FIG. 1B, a plot 110of a first property versus a second property (e.g., a property space)includes data within a first class and data within a second class. Onedirection 107 in the property space is associated with an analysis thataims to maximize variance between classes and minimize variance withinclasses while another direction 109 is associated with an analysis thataims to maximize overall variance. In the second scenario, therelationship between the two classes differs from that of the firstscenario. In particular, the data of class 1 does not overlap with thedata of class 2. Consequently, a projection of the data in the propertyspace onto the direction 107 allows for identification of two distinctclasses of data. In contrast, a projection of the data in the propertyspace onto the direction 109 provides little information to aid inclassification of the data into two distinct classes.

Thus, the scenario of plot 110 demonstrates an instance where ananalysis that seeks to maximize variance between classes and to minimizevariance within classes may provide useful information for classifyingdata in a property space. Further, the scenario of plot 110 demonstratesan instance where an analysis that seeks to maximize overall variancemay be limited in usefulness.

FIGS. 2A and 2B show additional exemplary concepts graphically. Aproperty space plot 205, as illustrated in FIG. 2A, includes data and adirection resulting from an analysis that aims to maximize between classvariance while minimizing within class variance. The data are projectedonto this direction to generate various points and a reference ischosen. Measurement of distances then occurs between the reference andthe various points. A plot 210 of distance versus counts (e.g., ahistogram), as illustrated in FIG. 2B, is formed by binning the measureddistances. A plot such as the plot 210 may be used in classifying thedata. For example, in the plot 205, a general observation of the datamay not readily discern whether classes exist; however, upon formationof a distance versus count plot or histogram, the existence of classesmay readily be discerned. In the example of FIG. 2B, the plot 210indicates that two classes appear to exist. Further, the plot 210 may bedeemed one histogram or two histograms, for example, one for each class.

The plots of FIGS. 1A, 1B, 2A and 2B illustrate manners in which dataand/or class regions (e.g., class properties) may be analyzed. Inparticular, analyses such as linear discriminant analysis and principlecomponent analysis may be used with respect to class properties and/ordata. As already mentioned, linear discriminant analysis aims tomaximize between class variance while minimizing within-class varianceand principle component analysis aims to maximize overall variance.Further details of such analyses appear below.

Various exemplary methods, devices, systems, etc., described hereinoptionally include projecting data onto a direction in a property space,measuring distances for projected data and binning distances. Suchbinned information may allow for identification of classes and classmembers (i.e., class data). Prior to discussion of further details ofvarious exemplary methods, devices, systems, etc., an overview ofmathematics germane to various analyses is presented.

Mathematical Details of Analyses

Definition of Variables

Two analyses that may be used in various exemplary methods, devices,systems, etc., are Fisher linear discriminant analysis and principlecomponent analysis. In order to describe the mathematics of Fisherlinear discriminant analysis and of principal component analysis it isconvenient to first define certain variables as follows.

Let the training set data for the classification be represented byx_(i)εR^(d), i =1, . . . , M. Each datum x_(i) in the set is associatedwith a column feature vector x_(i)=(x_(i1), x_(i2), . . . , x_(id))^(T),in which each component x_(ik) represents the kth property or attributeof the ith datum x_(i).

Let M be the total number of sample data, where each datum is indexed byi=1,2, . . . M. For example, if the training set comprised image pixelsof different colors in RGB color space, there would be M pixelsaltogether in the training set, each represented by an RGB color value.The dimensionality of the property space is d=3 and each pixel i isrepresented as a 3D color vector, where x_(i1)≡red value of the pixel,x_(i2)≡green value of the pixel, X_(i3)≡blue value of the pixel.

Let the sample data be divided into classes X_(j) indexed by j of whichthere are a total of N classes, i.e. j=1,2, . . . N. Within each class jlet the number of samples or data be m_(j), which implies that:

${\sum\limits_{j = 1}^{N}m_{j}} = {M.}$

Let μ_(j=(μ) _(j1), μ_(j2), . . . , μ_(jd))^(T) be the mean vector ofclass j, such that:

${\mu_{j} = {\frac{1}{m_{j}}{\sum\limits_{x_{i} \in X_{j}}x_{t}}}},{j = 1},\ldots\mspace{14mu},N$and let μ be the total mean vector for all classes taken together suchthat:

$\mu = {\frac{1}{M}{\sum\limits_{j = 1}^{N}{m_{j}{\mu_{j}.}}}}$

Define S_(j) to be the scatter matrix of class j, i.e.

$S_{j} = {\sum\limits_{x_{t} \in X_{j}}{\left( {x_{t} - \mu_{j}} \right) \cdot {\left( {x_{t} - \mu_{j}} \right)^{T}.}}}$

The matrix S_(j) is proportional to the corresponding covariance matrixK_(j) of class j, i.e. S_(j)=m_(j)·K_(j), so that:

$K_{j} = {\frac{1}{m_{j}}{\sum\limits_{x_{t} \in X_{j}}{\left( {x_{t} - \mu_{j}} \right) \cdot {\left( {x_{t} - \mu_{j}} \right)^{T}.}}}}$

Similarly let K be the covariance matrix of all samples (entire trainingdata set), i.e.

$K = {\frac{1}{M}{\sum\limits_{t = 1}^{M}{\left( {x_{t} - \mu} \right) \cdot {\left( {x_{t} - \mu} \right)^{T}.}}}}$

Additionally define matrix S_(W) as:

$S_{W} = {\sum\limits_{j = 1}^{N}S_{j}}$and let S_(B) be the scatter matrix of the class centers, i.e.

$S_{B} = {\sum\limits_{j = 1}^{N}{{m_{j}\left( {\mu_{j} - \mu} \right)} \cdot {\left( {\mu_{j} - \mu} \right)^{T}.}}}$

With the variables defined we can proceed to a description of analyticalmethods. One objective of linear discriminant analysis is to performdimensionality reduction while preserving as much of the classdiscriminatory information as possible by finding directions in propertyspace along which classes are best separated. This is done bysimultaneously considering the scatter within classes (as represented bythe matrix S_(W)) and between classes (as represented by matrix S_(B)).In contrast, principal component analysis has as an objective thereduction of dimensionality space by finding directions in propertyspace that maximally account for the variation in the overall sampledata without regard to class membership. This is done by using theoverall sample covariance (represented by matrix K) as a basis foranalysis.

Linear Discriminant Analysis

One of the goals of linear discriminant analysis (LDA) is to find aprojection w from R^(d) original property space (x_(i)εR^(d)) to asub-space of reduced dimensionality R^(N−1) by constructing N−1separating hyperplanes, such that the projection of the class centersμ_(j) onto the lower dimension space has the maximum variance (i.e., thecenters are as separated from each other as possible) and such that theprojections of the points x_(i)εX_(j) are clustered as close as possibleto the projections of the centers μ_(j). This goal can be formalizedinto the following optimization functional:

${{J(w)} = \left. \frac{{w^{T}S_{B}w}}{{w^{T}S_{W}w}}\Rightarrow\max\limits_{w} \right.},$where vertical bars |·| denote the determinant of a matrix. Finding thed ×(N−1) rectangular matrix w maximizing J(w) is equivalent to thesolution of the following generalized eigenvalue problem:S_(B)w=λS_(W)w.

Here λ is a real number known as the eigenvalue, which representsdiscrimination power. Each eigenvalue is associated with a column ofmatrix w, which is known as the eigenvector and represents thecontribution of each property to the eigenvalue. The eigenvalues λ arefound as the roots of the characteristic equation:|S _(B) −λ·S _(W)|=0.

After sorting the solutions λ in order of decreasing magnitude, i.e.,λ₁≧λ₂≧ . . . ≧λ_(N−1) one can then determine the correspondingeigenvectors w_(i) by solving(S _(B)−λ_(i) ·S _(W))·w _(i)=0, i=1, . . . , N−1.Principal Component Analysis

One of the goals of principal component analysis (PCA) is a variablereduction procedure typically resulting in a relatively small number ofcomponents that account for most of the variance in a set of observedvariables. The reduction is achieved by the linear transformation of theoriginal coordinates x=(x₁, x₂, . . . , x_(d))^(T) in R^(d) space intonew series of lower dimension principal components, say z=(z₁, . . . ,z_(p)) in R^(p)(p<d) defined as

${z_{i} = {\sum\limits_{k = 1}^{d}{U_{ik}x_{k}}}},{i = 1},\ldots\mspace{14mu},{p;{p < d};}$that are orthogonal (i.e., uncorrelated). Here U_(ik) is k-th componentof i-th eigenvector U_(i)=(U_(il), . . . , U_(ik), . . . , U_(id))^(T)of the covariance matrix, K, of all samples, i.e.KU_(i)=μ_(i)U_(i),where μ_(i), is the corresponding eigenvalue.

Geometrically, principal component analysis constructs a set ofuncorrelated directions that are ordered by their variance. Thus, thefirst principal component z_(l) is usually considered to have themaximum variance and explains the largest percentage of the totalvariance, the second PC explains the largest percentage of the remainingvariance, and so on. Maximizing the variance of the first component isequivalent to the maximization of:U₁ ^(T)KU₁.

Taking into account the normalization condition:U₁ ^(T)U₁=1.U₁=(U₁, . . . , U_(1d)) is the first eigenvector of the matrix Kassociated with the maximal eigenvalue μ_(i), i.e. KU₁=μ₁U₁. Eachsubsequent component z_(i) (i=2, . . . , p) contains the maximumvariance for any axes orthogonal to the previous component and theeigenvector U_(i) corresponding to it is defined as:KU_(i)=μ_(i)U_(i), i=2, . . . p; μ₁≧μ₂≧ . . . ≧μ_(p).Thus, the (U₁, . . . , U_(p)) are defined as the p leading eigenvectorsof K. The eigenvalue associated with each vector is the variance in thatdirection. For Gaussian data the principal components are the axes of anequiprobability ellipsoid.

In some situations directions with the most variance are especiallyrelevant to clustering. However, since principal component analysiscares only about the scatter of the entire data set, the projection axeschosen might not provide good discrimination power. Nevertheless, thedistribution of the data in the space of principal components differsfrom that in the original property space, whereas linear discriminantanalysis merely partitions the original property space in a way thatmaximizes class separation. Accordingly, there can be advantages to theuse of principal component analysis in conjunction with lineardiscriminant analysis.

In one exemplary implementation, an iterative classifier is providedbased on linear Fisher discriminant analysis. According to such aclassifier, property distances are optionally projected onto a vector toform property histograms. Such histograms may be subjected to nestinganalysis, which may include objectives of, for example, identifying andeliminating correctly classified data. Where data elimination occurs,remaining data may be subjected to reclassification using linear Fisherdiscriminant analysis and/or other analyses. An exemplary methodoptionally includes such a process wherein iterations occur until fullclassification is achieved or no further simplification is possible.

The aforementioned mathematics aim to explain some basics of Fisherlinear discriminant analysis and other analyses that generally pertainto variance of data.

Histograms, Formation and Classification

Various exemplary methods, devices, systems, etc., include formation ofhistograms and histogram nesting analysis. For ease of explanation suchconcepts will initially be presented for the case of two classesassociated with multiple properties. Subsequently, the description willbe extended to more than two classes.

Consider the eigenvector resulting from linear discriminant analysis, ofwhich there is only one in the two-class case. This vector or projectionvector provides a direction in sample property space. A reference planenormal to the vector may be constructed at any convenient location alongthe vector, for example through the origin of property space. For everydatum a distance from the plane parallel to the vector direction may becomputed and a histogram of these distances constructed. This is ahistogram of the projection of the data onto the vector. This histogramcontains distances for data that are part of the training set, which islabeled according to class membership. Accordingly, each distance in thehistogram may be labeled by the class of the datum projected.Conceptually, this may be considered two separate histograms, one forthe first class and one for the second class. In addition, variousmanners exist in arriving at such histograms, for example, use of aplane normal to a projection vector, selection of a reference point andprojection of data onto a vector, etc.

These two conceptual class histograms may have four relationships:separated, overlapped, enclosed and matched, as shown in FIG. 3, inwhich the width of each box represents the range of data enclosed withinthe histogram.

Nesting analysis of the histograms comprises considering these differentcases and involves assignment of data to a class where possible,retaining unseparated or unclassified data for further analysis. In theseparated case, the histograms of the two classes do not overlap andperfect classification of the training set data is achieved along theprojection vector.

In some instances, a decision may be made about how to test data fallingbetween the two class histograms including determining whether theinstance may signal an unrepresentative training set. Such an instancemay be addressed, for example, by using fitted distributions, includingmixture models, in conjunction with Bayesian methods, or by partitioningthe gap between histograms equally between the two classes.

In the overlapped case, class histograms overlap partially. Data outsidethe overlap region, which are indicated by shading, may be classifiedunambiguously. Two dividing planes normal to the projection vector andpositioned at boundaries of the overlap region may be used to classifytest data. Only the data in the histogram overlap region (i.e. betweenthe dividing planes) are retained for a further iteration of analysis(e.g., LDA, PCA, etc.).

The contained case is rather similar, and one class histogram iscompletely contained within the other. This may occur as illustrated, inwhich case two dividing planes are again constructed to bound theoverlap region. Alternatively, one or other end of the first classhistogram may align with an end of the second class histogram. In thiscase only a single dividing plane is required, with the other databoundary being formed by the common histogram limits. Again, data in theoverlap region are retained for another iteration of linear discriminantanalysis or other analysis while data outside this interval are labeledwith class membership.

The matched case arises when both class histograms have the same limitsand range along the projection vector. This situation may be treated inseveral ways. For example, a class membership probability may beassigned by Bayesian techniques or by minimization of totalclassification error. In these cases complete classification isachieved. Alternatively, multiple dividing planes may be constructedwithin the histograms to enclose and classify regions that are presentonly in a single histogram. Remaining regions that are not classifiedare optionally retained for a further iteration of linear discriminantanalysis or other analysis, which is illustrated in FIG. 4.

In FIG. 4, a region marked A in the Class 1 histogram has nocorresponding distances in the Class 2 histogram. It may, therefore, beassigned to class 1. Similarly, the peaks marked B in the Class 2histogram have no corresponding distances in the Class 1 histogram andmay be assigned to class 2. In contrast, the peaks marked C and C′ arecommon to both histograms and are retained for further iterations oflinear discriminant analysis or other analysis.

Various exemplary methods, devices, systems, etc., optionally excludedata that have been classified in the course of nesting analysis fromfurther consideration since they have already been assigned a classmembership. In such instances, remaining data may be subject to anotherlinear discriminant analysis or other analysis to receive a new vectorand new projected histograms. The cycle of nesting analysis and analysismay be repeated until all the data have been classified.

By means of histogram nesting analysis a series of planes dividingproperty space may be constructed to supplement the planes (or plane inthe case of binary classification) provided by vector analysis. By thesemeans a more effective classification may be obtained.

While the preceding discussion was generally couched in terms ofhistograms derived from projection onto a vector stemming from ananalysis such as linear discriminant analysis, projection onto a vectorstemming from a principal component analysis may be used (e.g., theleading principle component vector, etc.). The latter approach isoptionally beneficial for multimodal distributions such as thoseillustrated in FIG. 5, which for simplicity of illustration involves twoclasses each associated with two class properties.

As noted earlier, Fisher linear discriminant analysis is based on anassumed Gaussian distribution of classes and may be expected to havesome difficulty coping when the data do not match such assumptions. Ingeneral, multimodal distributions can diminish the classifying abilityof a classifier reliant on Fisher linear discriminant analysis.

FIG. 5 shows two classes wherein class 1 has a significantlynon-Gaussian distribution (e.g., multimodal, etc.). Projection of theclass property vectors onto the leading linear discriminant analysisvector creates distributions of distances which overlap for the twoclasses. In contrast, projection of properties onto the leadingprincipal component vector creates class histograms of the containedtype where the histogram for class 2 lies within the histogram for class1. Dividing planes along and normal to the principal component vectorimmediately lead to separation of the classes.

Accordingly, various exemplary methods, devices, systems, etc., includeconsideration of property projections on both a linear discriminantanalysis vector and a principal component analysis vector. Criteria forchoosing one projection over the other for nesting analysis may bechosen according to convenience, expedience, desired results, etc. Forexample, projections may be chosen that lead to classification of thelargest number of data. However, it is preferred to choose a projectionwhich leads to the classification of the largest range of the data ateach stage. This has the benefit of generally reducing the number ofdiscriminant analysis and projection iterations that must be performedto achieve complete classification. This objective may be achieved bychoosing the projection that minimizes overlap of class histograms whilemaximizing the width of the histograms, so maximizing the range ofprojected distances that can be assigned to a class.

A number of auxiliary criteria may be used depending on the nature ofthe data and the classification problem being solved. For example, thedistance between the means of the class histograms divided by the squareroot of the sum of the histogram variances along the discriminant vectormay be used as a test. For large values of this quantity (i.e. compactand well-separated distributions) nesting analysis may be based solelyon the linear discriminant analysis vector projection.

For small values of the metric, an optimal projection vector may bechosen by testing or assessing both the linear discriminant analysisvector and the principal component analysis vector projections.

Other criteria may be used as well. For instance, other criteria mayinvolve metrics based on histogram width ranges or histogram medianvalues. In general the metrics may be chosen to minimize the overallamount of computation required to achieve classification by striking anoptimal balance between effort or resources expended on nesting analysisrelative to the effort or resources expended on multiple iterations ofthe process.

In the case of more than two classes the nesting analysis becomes morecomplex than that described for two classes. For more than two classesnot only are there more pairwise relationships between classes toconsider but there are also multiple discriminant vectors to consider,one fewer than the number of classes.

For N classes and a projection of class properties onto a single vector,the histogram of the first class must be compared with the histogramsfor N−1 remaining classes. The histogram of the second class must becompared with N−2 remaining class histograms, the histogram of the thirdclass with N−3 remaining histograms, and so on. The total number ofcomparisons for a single projection is thus 0.5 N(N−1). If thesecomparisons are performed for each of the N−1 vectors the total numberof pairwise histogram comparisons becomes 0.5 N(N−1)².

While such comparisons are realistic for a relatively small number ofclasses such as three or four, the comparisons rapidly become unwieldyfor a larger number of classes. However, considerable simplification ispossible. For example, only the leading linear discriminant analysisvector, or possibly a leading few vectors, may be considered, optionallyalong with the leading principal component analysis vector. This isbecause the iterative nature of the classification does not requireoptimal classification from a given linear discriminant analysis in aseries of such analyses and the use of the principal component vectorcan result in large regions of property space being assigned to a classin situations where linear discriminant analysis gives poor classseparation.

The nesting analysis may be simplified further by treating it as atwo-class situation. First, the histogram of any class, j, that does notoverlap with the histogram of the union of remaining classes may be usedto assign membership of classj. Classj may then be excluded from furtheranalysis. Second, the width range of each class histogram may becompared to the sum of the widths of the remaining class histograms.Then the histogram with the largest width relative to the width of theunited histogram of the remaining classes may be chosen for assigningclasses. Alternatively, the histogram with the least overlap with theunited histogram of the remaining classes may be chosen for assigningclasses. The situation is illustrated in FIG. 6, which contains severalexamples of schematic histograms labeled according to the classes theycontain.

For the case of overlapping histograms, class i would be selected overclass j for partitioning property space, since a larger range ofproperty space may be assigned to class i than classj as illustrated bythe sizes of the shaded regions. However, both class i and classj couldalso be used in conjunction with each other.

For the case of contained histograms, class k would be selected overclass l for partitioning property space since a larger range of thisspace may be assigned to class k than to class l, though the classescould also be used in concert. Either class i in the case of overlap orclass k in the case of containment may be chosen to partition propertyspace depending on which class accounts for the largest fragment ofproperty space. Preferably both class i and class k are used together toreduce the volume of property space so as to limit what is used in thenext linear discriminant analysis iteration or other analysis iteration.

Additional possibilities are contemplated. For example, the range ofproperty space assigned to a class may be compared against a thresholdand, if it is insufficient, projection onto the leading principalcomponent may be used for histogram nesting analysis. Similarly, ifcomparison of every class histogram with the histogram for the union ofremaining classes leads to the matched histogram case, projection onto aprincipal component may be chosen. Alternatively, the next mostsignificant linear discriminant analysis vector may be included in thenesting analysis.

Exemplary histogram analyses have been illustrated herein with respectto certain representative examples. However, it should be appreciatedthat these representative examples are only exemplary and are notintended to be limiting. Other alternative implementations are possiblewithin the broad scope of nesting analysis if they are directed toreducing the number of data to be classified, or to improving thequality, accuracy or reliability of the partitioning of property space,or to increasing the efficiency or efficacy of classification.

Various Exemplary Methods

FIG. 7 shows an exemplary method 700 for classifying data in amulti-dimensional property space. The exemplary method 700 commences ina start block 704. Next, a determination block 708 determines aprojection vector that, for example, aims to maximize between classvariance and minimize within class variance of data. A projection block712 follows that projects the data onto the projection vector. Aclassification block 716 relies on the projection to classify the data,for example, using histograms and a nesting analysis. The classificationblock 716 optionally includes eliminating data to form a modified dataset. A decision block 720 follows that decides if further classificationis required, possible and/or desirable. If the decision block 720decides that no further classification need occur, then the exemplarymethod 700 terminates in an end block 724; however, upon a decision formore classification, the method 700 continues at the determination block708, optionally after elimination of classified data to form a modifieddata set.

While the exemplary method 700 is described above with respect to datain a property space where classes may not be know a priori, analternative example optionally uses property-based classes that areknown a priori for generation of information germane to classifying. Forexample, a property-based class region may be defined and subject to theexemplary method 700 or a training data set may include data generatedfrom known and perhaps well-defined property-based classes. In theseexamples, the classifying may be optimized to more readily identifyclasses and/or class data in a test data set.

FIG. 8 shows an exemplary method 800 for classifying data in amulti-dimensional property space. The exemplary method 800 commences ina start block 804. Next, a determination block 808 determines more thanone projection vector. Each projection vector may, for example, aim tomaximize between class variance and minimize within class variance ofdata or aim to maximize overall variance of data. A projection block 812follows that projects the data onto each of the projection vectors. Aselection block 816 selects the better or best projection for subsequentuse in the exemplary method 800 (e.g., according to one or morecriteria). A classification block 820 relies on the selected projectionto classify the data, for example, using histograms and a nestinganalysis. The classification block 820 optionally includes eliminatingdata to form a modified data set. A decision block 824 follows thatdecides if further classification is required, possible and/ordesirable. If the decision block 824 decides that no furtherclassification need occur, then the exemplary method 800 terminates inan end block 828; however, upon a decision for more classification, themethod 800 continues at the determination block 808, optionally afterelimination of classified data to form a modified data set.

While the exemplary method 800 is described above with respect to datain a property space where classes may not be know a priori, analternative example optionally uses property-based classes that areknown a priori for generation of information germane to classifying. Forexample, a property-based class region may be defined and subject to theexemplary method 800 or a training data set may include data generatedfrom known and perhaps well-defined property-based classes. In anotherexample, training data is selected based on observation and/or one ormore criteria to aid in defining property-based classes. In theseexamples, the classifying may be optimized to more readily identifyclasses and/or class data in a test data set (see, e.g., examples belowwherein selection of some image data may occur prior to classificationof other image data).

An exemplary method includes receiving an image data set that comprisesa multidimensional property space and data classifiable into dataclasses, determining a projection vector for data of the data setwherein the projection vector maximizes a ratio of between-class scatteror variance to within-class scatter or variance, selecting a referencefor the vector, projecting at least some of the data onto the vector,measuring distances from the reference to at least some of the data,classifying at least some of the data into data classes based on anesting analysis of the distances, eliminating the classified data fromthe image data set to produce a modified image data set and decidingwhether to determine another projection vector for the modified imagedata set. While this exemplary method includes projecting at least someof the data onto the vector, if, for example, a reference plane isselected then the measuring may measure distances substantially parallelto the vector to achieve the same result.

Various exemplary methods optionally include property space dimensionsthat include color property dimensions (e.g., a red property dimension,a green property dimension and a blue property dimension). Where colorproperty dimensions are included in a property space, one or more dataclasses are optionally defined in part through use of a color vector.

As already mentioned, a linear discriminant analysis and/or a principlecomponent analysis may be used to determine a projection vector orvectors where appropriate. In addition, in some instances, an analysismay result in a matrix that includes a plurality of vectors. Where aplurality of vectors occurs, the leading vector may have particularusefulness in classifying.

Various exemplary methods optionally rely on eigen analysis whereineigenvalues and/or eigenvectors are involved, for example, determining aprojection vector may include determining one or more eigenvalues and/orone or more eigenvectors.

Various exemplary methods determine a projection vector that maximizesvariance between means of data classes wherein the means of data classesoptionally are representable via mean vectors.

Various exemplary methods include selection of a reference wherein thereference is optionally a point, a line, a plane, a hyperplane, etc.,the selection of which may depend on property space dimensions and/orreduction of property space dimension through one or more projectiontechniques. Where an exemplary method includes selection of a planenormal to a projection vector, measuring distances may occur fordistances parallel to the projection vector.

Various exemplary methods optionally include binning distances to formone or more histograms. Various exemplary methods optionally includenesting analysis that determines a relationship between two or morehistograms or regions within a single histogram. Such a relationship mayinclude separated, overlapped, enclosed and matched or anotherrelationship.

In instances where a modified image data set is created, an exemplarymethod optionally includes displaying the modified image data set on adisplay device. An exemplary method may optionally display classifieddata on a display device, if appropriate.

In deciding whether to continue with a subsequent iteration, anexemplary method may optionally consider classified data, unclassifieddata or classified data and unclassified data. A decision may includedetermining whether to perform a principle component analysis on atleast some of the data, determining whether to determine a projectionvector that maximizes overall variance of at least some of the data,and/or other determining. A decision may include consideration of rangeof data classified, consideration of width of a histogram formed bybinned distances, and/or one or more other considerations.

An exemplary method optionally includes displaying an image data set andselecting one or more regions of the image wherein the selecting acts todefine one or more data classes. In such instances, the selectingoptionally acts to select one or more regions to keep and one or moreregions to drop from the image.

An exemplary method includes receiving an image data set in amultidimensional property space that comprises data classifiable intodata classes, determining a projection vector that maximizes overallvariance of the data of the image data set, determining a projectionvector that maximizes variance between classes of the image data set andminimizes variance within classes of the image data set, deciding whichvector allows for distinguishing more classes, projecting at least someof the data onto the vector that allows for distinguishing more classes,classifying at least some of the data based on the projecting,eliminating at least some of the data from the image data set based onthe classifying to form a modified image data set and determining one ormore additional projection vectors selected from a group consisting ofprojection vectors that maximize overall variance and projection vectorsthat maximize variance between classes and minimize variance withinclasses.

Such an exemplary method optionally includes binning distances to formone or more histograms (e.g., as part of classifying). Classifying mayoptionally include nesting analysis or analyses. For example,classifying may include binning distances to form one or more histogramsand a nesting analysis of the one or more histograms.

Such an exemplary method optionally includes deciding whether todetermine another projection vector that maximizes variance betweenclasses of the modified image data set and minimizes variance withinclasses of the modified image data set and/or deciding whether todetermine another projection vector that maximizes overall variance.

Various exemplary methods are optionally performed using one or morecomputer-readable media that include instructions capable of executionin conjunction with a processor to perform at least some aspects of theexemplary method.

EXAMPLES

An example of practical utility of the various exemplary implementationsdescribed herein can be illustrated with a color classification problem,which starts with a color image in which each pixel is represented by acolor vector. The vector is three-dimensional, corresponding to a redchannel value, a green channel value, and a blue channel value. The taskis to mark some colors as desired and others as undesired and then toachieve classification of the image into wanted and unwanted elements.The unwanted elements are erased to transparency, leaving only thedesired elements visible in the image. Such a task may be accomplishedby manually erasing individual pixels so that only the desired onesremain. However, this is an extremely laborious process requiring muchdexterity and patience. This may be appreciated when it is borne in mindthat digital camera images, for instance, contain several millions ofpixels. Much time and effort could be saved if it were possible to marksome representative colors to keep and to discard and, on that basis,automatically erase what was undesired while retaining what was desired.

In FIGS. 9A-D, the original images are in color and represented hereinin grayscale. The description that follows refers to colors that may beappreciated to represent a normally exposed and normally processed colorimage of a woman. FIG. 9A shows an original image of a woman surroundedby a collection of color patches. It is desired to erase all the colorpatches and the background, leaving only the woman. Note that many ofthe red patches are very similar to the color tones in the skin of thewoman.

FIG. 9B shows in white regions of the image that were roughly markedwith a brush tool as containing colors that are to be kept. Shown inblack are the colors marked as those that are to be dropped or erased.The keep and drop colors form a training set for classifying theremaining pixels of the image into those that should be erased and thosethat should be retained.

FIG. 9C shows the result of conventional linear discriminant analysisapplied to this classification problem, where the checkerboard showsregions of the image that have become transparent. Red color patches arenot removed but portions of the woman's skin and hair disappear.Additionally the black patch is not removed and some of the greenpatches do not become completely transparent. This level ofclassification is not useful relative to the large amount of additionalmanual correction required for the image to meet the separationrequirements.

FIG. 9D shows the result of using an exemplary iterative lineardiscriminant analysis scheme in accordance with various exemplaryimplementations described herein, including the choice of projectiononto a linear discriminant analysis or principal component analysisvector. The results are objectively satisfactory. All the color patchesare removed and the woman is retrieved in her entirety, requiring onlyslight correction of three pixels on the woman's lip, two on the leftearlobe and two on the right eye.

In FIGS. 10A-C, the original images are in color and represented hereinin grayscale. The description that follows refers to colors that may beappreciated to represent a normally exposed and normally processed colorimage of flowers (e.g., a red rose and a blue and yellow iris).

FIGS. 10A-C show another example of color classification, involving animage of very poor quality containing numerous artifacts from excessiveJPEG compression (FIG. 10-A). Additionally, the colors of the flowersshow lightness and saturation variations as well as spill-over of colorfrom the JPEG compression. The task is to retain the blue and yellow andred flowers while removing the dark background, green leaves and whiteflowers.

FIG. 10A is the original image. FIG. 10B shows how keep and drop colorswere roughly marked in the image with a brush tool. FIG. 10C shows theresult of color classification using various exemplary methods describedherein. Despite the poor quality of the image a good separation of thedesired objects is obtained, requiring only a small amount of clean-up.

It will be appreciated that the various exemplary methods describedherein may be implemented, all or in part, as one or more computerprograms or modules that include computer-executable instructions and/orcomputer readable data. These computer programs may be stored orembodied in one or more types of computer-readable medium. As usedherein, a computer-readable medium may be any available medium that canstore and/or embody computer-executable instructions and that may beaccessed by a computer or computing process. As used herein, a computerprogram product comprises a computer program or module embodied in or ona computer-readable medium.

Shown below in FIG. 11 is one example of a computing system 1100 inwhich the various methods described herein may be implemented. In itsmost basic configuration, the computing system 1100 includes aprocessing unit 1102, an input/output (I/O) section 1102, and a mainmemory 1104, including volatile and/or non-volatile memory.Additionally, the computing system may include or have access to variousmass storage devices or systems 1106, including various removable and/ornon-removable mass storage devices. Examples of mass storage devicesmight be, without limitation, various magnetic, optical, and/ornon-volatile semiconductor memory, etc. In the case where the massstorage device comprises a number of storage devices, those devices maybe distributed, such as across a computer network.

The computing system 1100 may have connected hereto input devices, suchas a keyboard 1107, a mouse (not shown), various optical scanners orreaders, microphones, video cameras, or various other computer inputdevices. The computing system 1100 may also have various output devicesconnected thereto, such as display devices 1108, speakers, printers, orvarious other computer output devices. The various input and outputdevices may be connected to the computing system 1100 via the I/Osection 1102. Other aspects of the computing system 1100 may includeappropriate devices 1111 to establish network or communicationsconnections to other devices, computers, networks, servers, etc., usingeither wired or wireless computer-readable media, and using variouscommunications protocols. For example, the computing system 1100 isshown in FIG. 11 as being connected to a remote computing system 1120.

The computing system 1100 and the remote computing system 1120 may be apart of, or in communication with, computer networks 1112, such as WideArea Networks (WAN), Local Area Network (LANs), the Internet, or any ofvarious other computer networks.

Although various implementations set forth herein have been described inlanguage specific to structural features and/or methodological steps, itis to be understood that the invention defined in the appended claims isnot necessarily limited to the specific features or steps described.Rather, the specific features and steps are disclosed as representativeforms of implementing the claimed invention.

1. A method comprising: receiving an image data set in amultidimensional property space that comprises data classifiable intodata classes; determining a projection vector that maximizes overallvariance of the data of the image data set using a processor;determining a projection vector that maximizes variance between classesof the image data set and minimizes variance within classes of the imagedata set using the processor; deciding which vector allows fordistinguishing more classes; projecting at least some of the data ontothe decided vector; classifying at least some of the data into aplurality of unambiguously defined data classes based on the projectingoperation, using the processor; eliminating at least some of theclassified unambiguously defined data classes from the image data setbased on the classifying operation and using the processor to form amodified image data set containing only unclassified data; anddetermining one or more additional projection vectors selected from agroup consisting of projection vectors that maximize overall varianceand projection vectors that maximize variance between the unambiguousclasses and minimize variance within the unambiguous classes.
 2. Themethod of claim 1 wherein the classifying comprises binning distances toform one or more histograms.
 3. The method of claim 1 wherein theclassifying comprises nesting analysis.
 4. The method of claim 1 whereinthe classifying comprises binning distances to form one or morehistograms and a nesting analysis of the one or more histograms.
 5. Themethod of claim 1 further comprising deciding whether to determineanother projection vector that maximizes overall variance of themodified image data set.
 6. The method of claim 1 further comprisingdeciding whether to determine another projection vector that maximizesvariance between classes of the modified image data set and minimizesvariance within classes of the modified image data set.
 7. The method ofclaim 1 wherein the dimensions of the property space comprises colorproperty dimensions.
 8. The method of claim 1 wherein the dimensions ofthe property space comprise a red property dimension, a green propertydimension and a blue property dimension.
 9. The method of claim 1wherein the data classes comprise a data class defined by at least acolor vector.
 10. One or more computer-readable media storingcomputer-readable instructions executable by a processor for performinga method comprising: receiving an image data set in a multidimensionalproperty space that comprises data classifiable into data classes;determining a projection vector that maximizes overall variance of thedata of the image data set using a processor; determining a projectionvector that maximizes variance between classes of the image data set andminimizes variance within classes of the image data set using theprocessor; deciding which vector allows for distinguishing more classes;projecting at least some of the data onto the decided vector;classifying at least some of the data into a plurality of unambiguouslydefined data classes based on the projecting operation, using theprocessor; eliminating at least some of the classified unambiguouslydefined data classes from the image data set based on the classifyingoperation and using the processor to form a modified image data setcontaining only unclassified data; and determining one or moreadditional projection vectors selected from a group consisting ofprojection vectors that maximize overall variance and projection vectorsthat maximize variance between the unambiguous classes and minimizevariance within the unambiguous classes.
 11. The one or morecomputer-readable media of claim 10 wherein the classifying comprisesbinning distances to form one or more histograms.
 12. The one or morecomputer-readable media of claim 10 wherein the classifying comprisesnesting analysis.
 13. The one or more computer-readable media of claim10 wherein the classifying comprises binning distances to form one ormore histograms and a nesting analysis of the one or more histograms.14. The one or more computer-readable media of claim 10, the methodfurther comprising deciding whether to determine another projectionvector that maximizes overall variance of the modified image data set.15. The one or more computer-readable media of claim 10, the methodfurther comprising deciding whether to determine another projectionvector that maximizes variance between classes of the modified imagedata set and minimizes variance within classes of the modified imagedata set.
 16. The one or more computer-readable media of claim 10wherein the dimensions of the property space comprises color propertydimensions.
 17. The one or more computer-readable media of claim 10wherein the dimensions of the property space comprise a red propertydimension, a green property dimension and a blue property dimension. 18.The one or more computer-readable media of claim 10 wherein the dataclasses comprise a data class defined by at least a color vector.